In general, for any 2D distribution of mass, the moment of inertia
about an axis orthogonal to the plane of the mass equals the sum of
the moments of inertia about any two mutually orthogonal axes in the
plane of the mass intersecting the first axis. To see this, consider
an arbitrary mass element
having rectilinear coordinates
in the plane of the mass. (All three coordinate axes
intersect at a point in the mass-distribution plane.) Then its moment
of inertia about the axis orthogonal to the mass plane is
while its moment of inertia about coordinate axes
within the mass-plane are respectively
and
.
This, the perpendicular axis theorem is an immediate consequence of
the Pythagorean theorem for right triangles.